What are the effects of indirect facilitation on the dynamic of a dryland ecosystem?

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.title[
# What are the effects of indirect facilitation on the dynamic of a dryland ecosystem?
]
.subtitle[
## Research associate application
]
.author[
### <strong>Alain Danet</strong>, Florian D. Schneider, Fabien Anthelme, Sonia Kéfi
]
.author[
### Illinois State University
]
.author[
### <a href="https://alaindanet.github.io">https://alaindanet.github.io</a>
]
.date[
### Slides created via the R package <a href="https://github.com/yihui/xaringan">xaringan</a>
]

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# Dryland ecosystems

41% of Earth land surface, 2bn of people

.footnote[[UNEP-WCMC](https://www.fao.org/dryland-assessment/en/)]

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# Catastrophic shifts
<img src="data:image/png;base64,#fig_ymca/catastrophic_shift1.png" width="70%" style="display: block; margin: auto;" />

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# Catastrophic shifts
<img src="data:image/png;base64,#fig_ymca/catastrophic_shift2.png" width="70%" style="display: block; margin: auto;" />

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# Catastrophic shifts
<img src="data:image/png;base64,#fig_ymca/catastrophic_shift3.png" width="70%" style="display: block; margin: auto;" />

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# Catastrophic shifts
<img src="data:image/png;base64,#fig_ymca/catastrophic_shift4.png" width="70%" style="display: block; margin: auto;" />

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# Catastrophic shifts
<img src="data:image/png;base64,#fig_ymca/catastrophic_shift5.png" width="70%" style="display: block; margin: auto;" />

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# Catastrophic shifts
<img src="data:image/png;base64,#fig_ymca/catastrophic_shift6.png" width="70%" style="display: block; margin: auto;" />

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# Direct facilitation

.pull-left[

<img src="data:image/png;base64,#fig_ymca/goodpatch-badpatch.jpg" width="70%" style="display: block; margin: auto;" />
## Improving local conditions 
- Increasing resources 
- Limiting erosion
- Limiting evapotranspiration

]

.pull-right[

## Positive feedback loop

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# Overgrazing

.footnote[[IPCC](https://www.ipcc.ch/srccl/chapter/chapter-3/), [brittannica.com](http://media-3.web.britannica.com/eb-media/55/20155-004-C8550C5E.jpg)]

???
Bisigato and Laphitz (2009) identified overgrazing as a cause of desertification
in the Patagonian Monte region of Argentina

Recent human-induced turning points in ecosystem functioning were uncovered around Volgograd (Russia) and around Lake Balkhash (Kazakhstan), attributed to recultivation, increased salinisation, and increased grazing.

Successful grass establishment has been impeded by overgrazing and nutrient depletion leading to the encroachment of shrubs into the northern Chihuahuan Desert (USA) since the mid-19th century (Kidron and Gutschick 2017
).

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# Indirect facilitation
<img src="data:image/png;base64,#fig_ymca/indirect_facilitation1.png" width="70%" style="display: block; margin: auto;" />

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# Indirect facilitation
<img src="data:image/png;base64,#fig_ymca/indirect_facilitation2.png" width="70%" style="display: block; margin: auto;" />

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# Indirect facilitation
<img src="data:image/png;base64,#fig_ymca/indirect_facilitation3.png" width="70%" style="display: block; margin: auto;" />

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# Indirect facilitation
<img src="data:image/png;base64,#fig_ymca/indirect_facilitation4.png" width="70%" style="display: block; margin: auto;" />

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# Two different stress and two types of facilitation

What is the effect of indirect facilitation through grazing on the dynamic of a
dryland ecosystem?

Coexistence, Stability, Cooccurrence

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# Transposing a dryland ecosystem to a cellular automata

.pull-left[
<img src="data:image/png;base64,#fig_ymca/cyprus.jpg" width="70%" style="display: block; margin: auto;" />
]

.pull-right[
<img src="data:image/png;base64,#fig_ymca/random_landscape-eps-converted-to.png" width="70%" style="display: block; margin: auto;" />
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# Cells

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# Transition equations

- Mortality:

$$
`\begin{aligned}
w_{n,o} =& m \\
w_{p,o} =& m 
\end{aligned}`
$$

- Degradation: 
$$
`\begin{aligned}
w_{o,-} =& d 
\end{aligned}`
$$

- Regeneration: 
$$
`\begin{aligned}
w_{-,o} =& r + f q_{+|-}
\end{aligned}`
$$

`$q_{i|j}$`:  proportion of i cells around j cells (local neighborhood)

---

# Colonization equations

- Nurse colonization:
`$$\begin{aligned}w_{o,n}=& (\red{\delta\rho_{n}} + (1-\delta)q_{n|o}) (b - c\rho_+ - \gamma)\end{aligned}$$`
- Protegee colonization:

`$$\begin{aligned}w_{o,p}=& (\red{\delta\rho_{p}} + (1-\delta)q_{p|o}) (b - c\rho_+ - g(1 - p))\end{aligned}$$`
Mecanism: .red[Global dispersal]

---

# Colonization equations

- Nurse colonization:
`$$\begin{aligned}w_{o,n}=& (\delta\rho_{n} + \red{(1-\delta)q_{n|o}}) (b - c\rho_+ - \gamma)\end{aligned}$$`
- Protegee colonization:

`$$\begin{aligned}w_{o,p}=& (\delta\rho_{p} + \red{(1-\delta)q_{p|o}}) (b - c\rho_+ - g(1 - p))\end{aligned}$$`
Mecanism: .red[Local dispersal]

---

# Colonization equations

- Nurse colonization:
`$$\begin{aligned}w_{o,n}=& (\delta\rho_{n} + (1-\delta)q_{n|o}) (\red{b} - c\rho_+ - \gamma)\end{aligned}$$`
- Protegee colonization:

`$$\begin{aligned}w_{o,p}=& (\delta\rho_{p} + (1-\delta)q_{p|o}) (\red{b} - c\rho_+ - g(1 - p))\end{aligned}$$`
Mecanism: .red[Maximum survival and germination]

---

# Colonization equations

- Nurse colonization:
`$$\begin{aligned}w_{o,n}=& (\delta\rho_{n} + (1-\delta)q_{n|o}) (b - \red{c\rho_+} - \gamma)\end{aligned}$$`
- Protegee colonization:

`$$\begin{aligned}w_{o,p}=& (\delta\rho_{p} + (1-\delta)q_{p|o}) (b - \red{c\rho_+} - g(1 - p))\end{aligned}$$`
Mecanism: .red[Global competition for resources]

---

# Colonization equations

- Nurse colonization:
`$$\begin{aligned}w_{o,n}=& (\delta\rho_{n} + (1-\delta)q_{n|o}) (b - c\rho_+ - \red{\gamma})\end{aligned}$$`
- Protegee colonization:

`$$\begin{aligned}w_{o,p}=& (\delta\rho_{p} + (1-\delta)q_{p|o}) (b - c\rho_+ - g(1 - p))\end{aligned}$$`
Mecanism: .red[Cost of defense against grazing]

---

# Colonization equations

- Nurse colonization:
`$$\begin{aligned}w_{o,n}=& (\delta\rho_{n} + (1-\delta)q_{n|o}) (b - c\rho_+ - \gamma)\end{aligned}$$`
- Protegee colonization:

`$$\begin{aligned}w_{o,p}=& (\delta\rho_{p} + (1-\delta)q_{p|o}) (b - c\rho_+ - \red{g(1 - p)})\end{aligned}$$`
Mecanism: .red[Grazing and indirect facilitation]

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# Pair approximation
<img src="data:image/png;base64,#fig_ymca/pair-approx1.png" width="70%" style="display: block; margin: auto;" />

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# Pair approximation
<img src="data:image/png;base64,#fig_ymca/pair-approx2.png" width="70%" style="display: block; margin: auto;" />

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# Question

How does indirect facilitation affect coexistence between two species in
competition?

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# Coexistence

Indirect facilitation allows coexistence between two species in competition

.pull-left[
<img src="data:image/png;base64,#fig_ymca/stability_u0.png" width="90%" style="display: block; margin: auto;" />
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.pull-right[
<img src="data:image/png;base64,#fig_ymca/stability_u5.png" width="90%" style="display: block; margin: auto;" />
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# More about coexistence

.pull-left[

## Classic apparent competition

Herbivore maintains coexistence

]

.pull-right[

## Indirect facilitation

Nurse maintains coexistence

]

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# Question

How does indirect facilitation affect stability along the aridity gradient?

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# Stability

.pull-left[

## Without indirect facilitation

<img src="data:image/png;base64,#fig_ymca/bifurc_g=0.1_u=5.png" width="100%" style="display: block; margin: auto;" />
]

.pull-right[

## With indirect facilitation

<img src="data:image/png;base64,#fig_ymca/bifurc_g=0.25_u=5.png" width="100%" style="display: block; margin: auto;" />
]

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# Question

How does indirect facilitation affect cooccurrence between the nurse and the
protegee?

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# Cooccurrence

.pull-left[

## Clustering nurse protegee 
<img src="data:image/png;base64,#fig_ymca/clustering_u0_u10.png" width="90%" style="display: block; margin: auto;" />
]

.pull-right[
## Nb of protegee | nurse
<img src="data:image/png;base64,#fig_ymca/protegee_number_u0_u10.png" width="100%" style="display: block; margin: auto;" />
]

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# Take home message

## Indirect facilitation ...

allows coexistence between two species in competition

modifies the transition of the protegee along the aridity gradient: from
catastrophic to continuous & hysteresis (less predictable)

does not necessarily affect species cooccurrence

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# Thanks